What is Phase portrait? — Physics

Definition 13.14 University Physics — Year 1 · Chapter 13 — Work, Energy, and Potential Energy

The phase portrait of a one-dimensional motion is the family of curves traced by the point $(x, \dot x)$ in the phase plane. For a conservative system each curve is a level line $\tfrac12 m\dot x^2 + E_p(x) = E_m$ : closed curves around stable equilibria (oscillations), open curves for free motion, and, through unstable equilibria, the separatrices that divide the two. Curves are traversed clockwise ( $\dot x > 0$ means $x$ increasing) and never cross.

Phase portrait of the simple pendulum, 1/2 2 + _02(1 - ) = const. Closed curves (blue): swings about the stable equilibrium = 0; open curves (orange): full rotations; between them the separatrix (red) through the unstable equilibria = ±π, the motion that just reaches the top. — Phase portrait of the simple pendulum, $\tfrac12\dot\theta^2 + \omega_0^2(1 - \cos\theta) = \text{const}$ . Closed curves (blue): swings about the stable equilibrium $\theta = 0$ ; open curves (orange): full rotations; between them the separatrix (red) through the unstable equilibria $\theta = \pm\pi$ , the motion that just reaches the top.

Examples

Example 13.15 (The pendulum’s landscape)

$E_p = -mg\ell\cos\theta$ : minimum at $\theta = 0$ , $E_p'' = mg\ell$ , so $\omega_0^2 = g\ell/(m\ell^2) = g/\ell$ — the small-oscillation result of Example 12.12 read off the well’s curvature; maximum at $\theta = \pi$ , the inverted pendulum, unstable. Energy $E_m < mg\ell$ : oscillation between $\pm\theta_{\max}$ ; $E_m > mg\ell$ : it goes over the top and rotates; $E_m = mg\ell$ : the separatrix, an infinitely slow approach to the top.

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Examples

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